On approximation of affine Baire-one functions View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

2003-12

AUTHORS

J. Lukeš, J. Malý, I. Netuka, M. Smrčka, J. Spurný

ABSTRACT

It is known (G. Choquet, G. Mokobodzki) that a Baire-one affine function on a compact convex set satisfies the barycentric formula and can be expressed as a pointwise limit of a sequence of continuous affine functions. Moreover, the space of Baire-one affine functions is uniformly closed. The aim of this paper is to discuss to what extent analogous properties are true in the context of general function spaces. In particular, we investigate the function spaceH(U), consisting of the functions continuous on the closure of a bounded open setU⊂ℝm and harmonic onU, which has been extensively studied in potential theory. We demonstrate that the barycentric formula does not hold for the spaceB1b(H(U)) of bounded functions which are pointwise limits of functions from the spaceH(U) and thatB1b(H(U)) is not uniformly closed. On the other hand, every Baire-oneH(U)-affine function (in particular a solution of the generalized Dirichlet problem for continuous boundary data) is a pointwise limit of a bounded sequence of functions belonging toH(U). It turns out that such a situation always occurs for simplicial spaces whereas it is not the case for general function spaces. The paper provides several characterizations of those Baire-one functions which can be approximated pointwise by bounded sequences of elements of a given function space. More... »

PAGES

255-287

Journal

TITLE

Israel Journal of Mathematics

ISSUE

1

VOLUME

134

Author Affiliations

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/bf02787408

DOI

http://dx.doi.org/10.1007/bf02787408

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1043408536


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